Trapezoid Flexagon

sample trapezoid flexagons

These flexagons are made from isosceles trapezoids (or trapeziums) and have some interesting properties. You can treat them as a triangle hexaflexagon with the tips cut off, using the pinch flex to explore the faces. Doing this reveals sides with two different shapes, one of which looks like a ring. In this state, you can also do a pivot flex on the inside of the ring, adding some unique behavior.

Trapezoid flexagons are one type of flexagon made from quadrilaterals such as squares. The ones shown here are trapezoid hexaflexagons, with six faces per side.

7 sided trapezoid flexagon

This first template is for a 7 sided trapezoid flexagon that allows the pinch flex and pivot flex. Click on the following two thumbnails to get the full sized versions. Print them, cut them out and paste them back to back. Cut along all the solid black lines and pre-crease along the dotted gray lines. Start by folding adjacent 7's together. Continue by folding together adjacent 6's, 5's, 4's and 3's. Repeat this process three times so only the 1's and 2's are visible. Finish by taping together the adjacent edges of the first and last trapezoids in the template.

template for 7 sided trapezoid flexagon, side a
Fig 1. 7 sided trapezoid flexagon, side a

template for 7 sided trapezoid flexagon, side b
Fig 2. 7 sided trapezoid flexagon, side b

5 pivot trapezoid flexagon

This next template is for a trapezoid flexagon that allows a series of 5 pivot flexes. Click on the following two thumbnails to get the full sized versions. Print them, cut them out and paste them back to back, with the trapezoids that have stars next to them back-to-back (so 8 and 7 back-to-back in the left ring and 3 and the bottom 2 back-to-back in the right ring). Cut along all the solid black lines and pre-crease along the dotted gray lines. Tape the starred edges together, making an overlapping spiral of two rings. Start by folding the 8's together. Continue by folding together the 7's, 6's, 5's, 4's and 3's. Finish by taping together the adjacent edges of the first and last trapezoids in the template.

template for 5 pivot trapezoid flexagon, side a
Fig 3. 5 pivot trapezoid flexagon, side a

template for 5 pivot trapezoid flexagon, side b
Fig 4. 5 pivot trapezoid flexagon, side b

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